Table Of Content8thAIAA/NASA/USAF/ISSMO Symposium on Multidiscipline Analysis and Optimization
6-8 September 2000, Long beach, CA
Efficient Simulation of Wing Modal Response: Application of
2 nd
Order Shape Sensitivities and Neural Networks
Rakesh K. Kapania* and Youhua Liu**
Virginia Polytechnic Institute and State University
Blacksburg, VA 24061
Abstract
At the preliminary design stage of a wing structure, an efficient simulation, one needing little
computation but yielding adequately accurate results for various response quantities, is essential in
the search of optimal design in a vast design space. In the present paper, methods of using
sensitivities up to 2"dorder, and direct application of neural networks are explored. The example
problem is how to decide the natural frequencies of a wing given the shape variables of the
structure. It is shown that when sensitivities cannot be obtained analytically, the finite difference
approach is usually more reliable than a semi-analytical approach provided an appropriate step size
is used. The use of second order sensitivities is proved of being able to yield much better results
than the case where only the first order sensitivities are used. When neural networks are trained to
relate the wing natural frequencies to the shape variables, a negligible computation effort is needed
to accurately determine the natural frequencies of a new design.
*: Professor, Department of Aerospace and Ocean Engineering, Associate Fellow AIAA
**: Research Assistant, Department of Aerospace and Ocean Engineering, Student member AIAA
Introduction
The modal response of wing structures is very important for assessing their dynamic response
including dynamic aeroelastic instabilities. Moreover, in a recent study _an efficient structural
optimization approach was developed using structural modes to represent the static aeroelastic wing
response (both displacements and stresses).
Sensitivity techniques are frequently used in structural design practices for searching the
optimal solutions near a baseline design 2.3.The design parameters for wing structure include
sizing-type variables (skin thickness, spar or rib sectional area etc.), shape variables (the plan
surface dimensions and ratios), and topological variables (total spar or rib number, wing topology
arrangements etc.). Sensitivities to the shape variables are extremely important because of the
nonlinear dependence of stiffness and mass terms on the shape design variables as compared to the
linear dependence on the sizing-type design variables.
Kapania and coworkers have addressed the first order shape sensitivities of the modal response,
divergence and flutter speed, and divergence dynamic pressure of laminated, box-wing or general
trapezoidal wing composed of skins, spars and ribs using various approaches of determining the
response sensitivities 4-t0.
In the present paper, the natural frequencies of general trapezoidal wing structures are to be
approximated using shape sensitivities up to the 2noorder, and different approaches of computing
the derivatives are investigated. The baseline design and shape sensitivities are calculated based on
an equivalent plate-model analysis (EPA) method developed by Kapania and Liu _. For
comparison, an efficient method that employs the artificial neural networks to relate the natural
frequencies of a wing to its shape variables is also established. An example of a 4 3 full factorial
experimentadl esign,i.e.,4levelsin 3variablesi,streatedbythesemethodstodisplaytheir
respectivemerits.
ShaDe Sensitivities
For a trapezoidal wing, there are four major independent shape variables: 1) the sweep angle A,
2) the aspect ratioa , 3) the taper ratio z, and 4) the plan area A. All the other dimensions of the
wing plate configuration can be calculated using these parameters as follows:
s = ,f_, a = 2_/a(1 + r),b = 2s/a(1 + r) (1)
where s is the length of semi-span, a and b are the chord-length at wing tip and root
respectively, as shown in Fig. 1.
The sensitivities for the design parameters at a baseline design point indicate trends in the
response of the baseline design if the parameters are perturbed. Usually, only the first order
derivatives are used:
. , 2. . _r_'Of0. i "
f(x',x z,...,x )= f(Xo,X o, ..,xo)+ _= -_x -Xo) (2)
where _)f0 _ _._f is the sensitivity at the baseline point with respect to the i-th design
Oxi _x "1_4,_ ,...,x_)
parameter. For a more accurate approximation, we can use higher-order derivatives in the Taylor's
expression:
n
.. , 2 'o,X... "
f(x',x 2, ",x")= f(Xo,Xo," f(x _, ,x o)
i=1 (3)
÷1
+3 (x' - x;) f(Xo,' Xo_,'", xo)
i=1
where besides the first order derivatives, second order derivatives --_)2f° (i,j= l,...n) are also
3xiOx i
used.
Equivalent Plate Analysis (EPA) of Trapezoidal Wing Structures
In Kapania and Liu _, a general method is developed to analyze trapezoidal wing structures
composed of skins, spars and ribs. The method is based on the Reissner-Mindlin model, a First-
Order Shear Deformation Theory (FSDT). An equivalent plate model is based on the hypothesis
that the original complex built-up structures behave like a plate, a simplification to reduce the
computational effort to obtain the free vibration and static responses. Compared with the methods
presented earlier in Kapania and Lovejoy t2and Cortial 1_,the formulation in Kapania and Liu tt
entails no limitation with respect to the wing thickness distribution. As shown in Ref. 11, the
method shows a good performance for both static and vibration problems in comparison with the
FEA. Free vibration and static response are obtained using the Ritz method. The advantages of
using Ritz method are: (i) the ease of carrying out formulation and computer realization, and (ii) the
global nature of the solution, important for accurate determination of various stress components.
Due to its efficiency in determining the natural frequencies and mode shapes of wings, the
Equivalent Plate Analysis (EPA) mentioned above can be used to investigate the variation of modal
response, that is, to evaluate the sensitivities of the natural frequencies with respect to trapezoidal
wing structures shape changes. For determining the response of the baseline design, the EPA can
be used, or the FEA employing a commercial package such as MSC/NASTRAN can be used for
better accuracy.
4
A key problem that needs to be addressed before this evaluation can be made is mode tracking.
The natural frequencies given by an ordinary eigenvalue solver are usually ranked by magnitude
but not by the modal content. As design variables are perturbed, frequencies drift and mode
crossing may occur. An algorithm for mode tracking is needed to maintain the correspondence
between eigenpairs of the baseline and the perturbed design. Several methods for such purpose
have been given by Eldred et al for self-adjoint _4and nonself-adjoint 15eigenvalue problems.
In the present study, a simple yet effective method is used. In this method, any ordinary
eigenvalue solver can be used, and the modes of the baseline structure are chosen as the
benchmarks. By using the modal assurance criterion (MAC) defined as
T --
({¢j} {¢i})2
= (4)
MACj, ({¢j}T{¢j IT{if,})
where {4j } and {¢i} are the eigenvector for the perturbed and the baseline design respectively, if
MAC j, = max(MACti ), we say that the j -th mode of the perturbed design corresponds to the i-th
l
mode of the baseline structure.
Aporoaches to Sensitivity Evaluation
There can be three kinds of approach for obtaining sensitivity derivatives: the finite difference
approach, the analytical approach, and the semi-analytical approaches. The finite difference
approach is very simple to formulate and implement, but is numerically inefficient and is sensitive
to the step-size used. A too-large step size usually causes significant truncation errors and a too-
small step size may lead to large round-off errors. As a result, the more elegant and accurate
analytical approach is used if it does not involve complex mathematical derivation. But for most
practical problems, the derivation of analytical derivatives is too formidable to handle manually.
ThebasicideabehindtheAutomaticDifferentiation(AD) isto let a computer to perform such
extensive tasks. The advantage of AD is to avoid truncation errors. The method has found
applications in sensitivity evaluation 9j0. For the basic theory of AD one can consult Ref. 16, and
for the state-of-the-art of AD one can refer to Ref. 17. If an approach uses both analytical and
finite-difference solutions to obtain the derivative, then it can be called a semi-analytical one.
The finite difference approaches can be constructed using the following formulas:
f'(x) = f (x + Ax) - f (x- Ax) FO(Ax 2) (5)
2Ax
f'(x) = f(x + Ax) - 2f(x) + f(x - Ax) + O(Ax 2) (6)
Ax 2
where
Ax = e. x (7)
in which E is the relative step size, but herein it is simply called the step size. Eq. (5) can be
a2f° (i _ j)
applied twice for evaluating the mixed second order derivatives such as axia x j
The analytical approaches for shape sensitivities of modal response can be based on the
following equations
(8)
a2, r(a[K] a[M] )
where
(2,.-2j) [. _ _,,---_ ¢_,},j*:i
(10)
l r[a[Ml)
6
here 6 is the shape variable, 2iand {¢i } are the i -th eigenvalue and eigenvetor, and (¢, }is mass-
normalized such that {¢i }r [M]{¢i }-- I. Equations (8) and (9) were first derived by Wittrick 18and
Fox and Kapoor _9respectively. One can find more on this topic in Ref. 20.
a[K] a[M]
The major difficulty of applying Eqs. (8) and (9) lies in the calculation of -- and --
a6 a6
air]
For instance, consider _. According to Ref. 11, the stiffness matrix [K] is formulated as an
a6
integral
v (11)
where only the inner part [G] = I_=[T] r [D][T]Jidz isa function of the shape variables, and the
Gaussian quadrature is used to obtain the integration on _ and r/. Therefore,
, , ¢at<]clae
- L ' (12)
0[G]
in which _ can either be determined analytically or numerically.
Often people make use of the advantages of both the finite difference and analytical approaches
in different stages of obtaining some complicated sensitivities. While trying to use the analytical
approach as much as possible, in other parts of the process the finite difference is used, as in the
cases of Refs. 8 and 10. This kind of approach is usually called semi-analytical.
In summary, there are three approaches to calculate the first order modal sensitivities:
3[K] b[MI
(i) analytical approach: Eqs.(8)-(10) are used, and and are determined
36 36
analytically.
3[K] /)[M] .
and m m Eqs.(8)-(lO) are determined
(ii) semi-analytical approach: Sensitivities _'6 _6
numerically, that is, for the case of --
,Eq. (12) is used where _-_ is calculated using a finite
difference scheme.
(iii) finite difference approach: -- are determined using Eq. (5) directly.
For the second order sensitivities, there can still be three approaches as specified above. While
the formulation for the analytical approaches is becoming more complicated, a scheme as simple as
Eq. (6) can be used for the finite difference approach.
Application of Sensitivity Technique (ST) in Multi-variable Optimization
In a multi-variable case, the following formulation is used instead of Eq. (3):
(13)
R_(p)=R(p_)+(p_pi)r 3R 1[ _ ]2
where p = (v', v2,..., v" )r is an arbitrary point in the design space, Pi = (vI,v_,..., vn,)r is the i -
th node point in the design space, Ri(p) is the response at p estimated by using the response and
its sensitivities at Pi, R(pi ) is the response at the i -th node point p_, and
3p, •_' _ 3v2' °°°_ _v° .
P=Pi
Once there are enough estimates for the response at p using Eq. (13), a more accurate
evaluation of response at p can be determined using the following weighting procedure involving
the so-called exponentially decaying influence function 2_.
R(p) = _ wi(p)R i(p) (14)
i
where i ranges through the N wdesign points which are closest to p, and the weight coefficients
w, (p) are determined such that its sum is unity:
exp(-Clp - p, ) (15)
w, (p) = ',','_exp(-C p - Pi )
i
in which C is an empirical constant, and the distance between p and Pi is defined as
tP - Pi = (VJ -v/) 2 .It can be seen that _.wi(p)=l.
i
Application of Neural Networks (NN)
Artificial Neural Networks (ANN), or simply Neural Networks (NN) are computational systems
inspired by the biological brain in their structure, data processing and restoring method, and
learning ability. More specifically, a neural network is defined as a massively parallel distributed
processor that has a natural propensity for storing experiential knowledge and making it available
for future use by resembling the brain in two aspects: (i) Knowledge is acquired by the network
through a learning process; (ii) Inter-neuron connection strengths known as synaptic weights (or
simply weights) are used to store the knowledge 22.
The NN has the following properties: (i) Many of its kind are universal approximators, in the
sense that, given a dimension (number of hidden layers and neurons of each layer) large enough,
any continuous mapping can be realized; Therefore, (ii) a NN provides a general mechanism for
\
9
buildingmodelsfromdata,or gives a general means to set up input-output mapping (iii) The input
and output relationship of NN can be highly nonlinear; (iv) A NN is parallel in nature, and it can
make computation fast when executed in a parallel computer, though NN can be simulated
in ordinary computers in a sequential manner.
Major steps of utilizing NN include: (i) specifying the topology or the structural parameters
(number of layers, number of neurons in each layer, etc.) of the NN, (ii) training of the NN,
corresponding to the learning process of the brain, (iii) simulation, corresponding to the recalling
function of the brain.
In the present work MATLAB Neural Network Toolbox was used. Ref. 23 contains a summary
of applications of NN in structural engineering and details of how to make use of MATLAB Neural
Network Toolbox.
Generally speaking, there can be two directions to use NN for the efficient simulation of the
performances of wing structures as in the following:
(a) Direct Application
In this case, the input layer includes all the design variables of interest (for instance, the four
shape parameters of the wing plan-form). The output layer gives the desired structural
responses, such as natural frequencies etc. The EPA is being used as the training data generator,
though if necessary, results obtained using a FEM can also be used as the training data.
Preparation of training data is very important, and the training algorithm used also greatly
impacts the training process 23. Caution must be exerted in specifying the network parameters
and training criterion so that the results of the trained network would not oscillate around the
training data. The direct application is what we do in this paper.
10
